#283604

order	512
nilpotency class	3
isoclinism	isoclinic class of #283579
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

622 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,7)(3,9,11,6)(5,8,12,15)(10,16,13,14)(17,18,23,27,24,28,22,38)(19,29,31,25,20,32,34,41)(21,35,36,26,37,42,39,43)(30,40,46,47,33,48,45,44);(3,10)(5,12)(6,14)(8,15)(9,16)(11,13)(17,19,23,31,24,20,22,34)(18,25,27,32,28,41,38,29)(21,30,36,46,37,33,39,45)(26,40,42,47,43,48,35,44);(1,3,4,11)(2,6,7,9)(5,13,12,10)(8,16,15,14)(17,20)(18,25)(19,24)(21,33)(22,34)(23,31)(26,40)(27,29)(28,41)(30,37)(32,38)(35,47)(36,46)(39,45)(42,44)(43,48);(17,21,24,37)(18,26,28,43)(19,30,20,33)(22,39,23,36)(25,40,41,48)(27,42,38,35)(29,44,32,47)(31,46,34,45)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gi,f^c=fh,g^c=ghi,e^d=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 190800);